Ensembles for Surface Reconstruction Yunjin Lee POSTECH Mincheol Yoon POSTECH Seungyong Lee POSTECH Ioannis Ivrissimtzis MPI Informatik Hans-Peter Seidel MPI Informatik {jin,mincheol,leesy}@postech.ac.kr, {ivrissim,hpseidel}@mpi-sb.mpg.de Abstract In this paper, we demonstrate that a statistical tech- nique, called ensembles, can improve the performance of a deterministic surface reconstruction algorithm. The ensemble technique first randomly subsamples the input point cloud, obtaining different subsets of points. These subsets are then separately processed by a deterministic algorithm and, finally, the several different outputs are combined into a single one with better quality. We ex- periment with the state-of-the-art surface reconstruction algorithm, Multi-level Partition of Unity implicits [10], showing that the ensembles can successfully handle point sets with noise and outliers. Keywords: statistical approach, ensembles, robust sur- face reconstruction, MPU implicits. 1. Introduction Excellent techniques have been proposed to recon- struct a high quality surface from a point cloud with the control of error bound. However, most of them are ba- sically geometry based deterministic algorithms. For in- complete data with much noise and outliers, they may fail to generate a satisfactory surface, in which case only careful user intervention can improve the result. A large category of these deterministic surface recon- struction algorithms can directly process points without normals. α -shapes are used in [2], B-spline patches are fitted with detail displacement vectors in [8]. In [1], a sur- face is constructed from the Delaunay tetrahedrization. Another category of algorithms requires points with normals as input. In this case, an implicit function f : R 3 → R is fitted to the input data and then the surface is extracted as the zero level set of f . This volumetric ap- proach was pioneered in [6]. The current state of the art in this approach includes a radial basis function based tech- nique [3] and the Multi-level Partition of Unity (MPU) implicits [10] which uses quadrics. In this paper, we argue that through random subsam- pling of the input data set, we can put a deterministic al- gorithm into a statistics based probabilistic setting, im- proving this way its robustness in the presence of noises and outliers. First, we randomly subsample the input data set to create several subsets. These subsets may overlap. Then we process each one of them separately, with the deterministic algorithm, obtaining several different out- puts. That is, we create an ensemble. Finally, we combine the members of the ensemble, i.e., the different outputs, to obtain a single output with better properties. The deterministic algorithm we use for experiments is the MPU implicits [10], which is suitable for reconstruct- ing a surface from large scattered point data equipped with normals. 2. Ensembles for Surface Reconstruction The input of the surface reconstruction ensemble is a point cloud P with normals and the output is a sur- face S. The algorithm is summarized with the following pseudocode; Surface Reconstruction Ensemble Input: Point cloud P with normals. Output: Surface S. 1. Create several overlapping random subsets of P. 2. Process the subsets separately reconstructing a surface for each subset. 3. Combine the reconstructions obtained in Step 2 into a single surface S. 2.1. Ensemble generation To obtain m subsets of P we simply use random sam- pling. In our experiments, we usually sample eleven sub- sets (m = 11) from P, each of which contains 10% of the points in P. Then, we use the MPU algorithm to gener- ate an implicit surface representation for each subset, ob- taining a set of m functions f j : R 3 → R, j = 1,..., m. (1) 2.2. Surface Averaging The zero level set of each function f j in Eq. (1) de- fines a surface S j . To obtain the ensemble surface S, we need to average these functions into a single function f whose zero level set will define S.
In this paper, we demonstrate that a statistical tech-nique, calledensembles, can improve the performanceof a deterministic surface reconstruction algorithm. Theensemble technique first randomly subsamples the inputpoint cloud, obtaining different subsets of points. Thesesubsets are then separately processed by a deterministicalgorithm and, finally, the several different outputs arecombined into a single one with better quality. We ex-periment with the state-of-the-art surface reconstructionalgorithm, Multi-level Partition of Unity implicits [10],showing that the ensembles can successfully handle pointsets with noise and outliers.
Excellent techniques have been proposed to recon-struct a high quality surface from a point cloud with thecontrol of error bound. However, most of them are ba-sically geometry based deterministic algorithms. For in-complete data with much noise and outliers, they mayfail to generate a satisfactory surface, in which case onlycareful user intervention can improve the result.
A large category of these deterministic surface recon-struction algorithms can directly process points withoutnormals.α-shapes are used in [2], B-spline patches arefitted with detail displacement vectors in [8]. In [1], a sur-face is constructed from the Delaunay tetrahedrization.
Another category of algorithms requires points withnormals as input. In this case, an implicit functionf :R3 → R is fitted to the input data and then the surfaceis extracted as the zero level set off . This volumetric ap-proach was pioneered in [6]. The current state of the art inthis approach includes a radial basis function based tech-nique [3] and the Multi-level Partition of Unity (MPU)implicits [10] which uses quadrics.
In this paper, we argue that through random subsam-pling of the input data set,we can put a deterministic al-gorithm into a statistics based probabilistic setting, im-proving this way its robustness in the presence of noisesand outliers. First, we randomly subsample the input data
set to create several subsets. These subsets may overlap.Then we process each one of them separately, with thedeterministic algorithm, obtaining several different out-puts. That is, we create anensemble. Finally, we combinethe members of the ensemble, i.e., the different outputs,to obtain a single output with better properties.
The deterministic algorithm we use for experiments isthe MPU implicits [10], which is suitable for reconstruct-ing a surface from large scattered point data equippedwith normals.
2. Ensembles for Surface Reconstruction
The input of the surface reconstruction ensemble isa point cloudP with normals and the output is a sur-faceS. The algorithm is summarized with the followingpseudocode;
Surface Reconstruction Ensemble
Input: Point cloudP with normals.Output:SurfaceS.
1. Create several overlapping random subsets ofP.2. Process the subsets separately reconstructing a surface
for each subset.3. Combine the reconstructions obtained in Step 2 into a
single surfaceS.
2.1. Ensemble generation
To obtainm subsets ofP we simply use random sam-pling. In our experiments, we usually sample eleven sub-sets(m= 11) from P, each of which contains 10% of thepoints inP. Then, we use the MPU algorithm to gener-ate an implicit surface representation for each subset, ob-taining a set ofm functions
f j : R3 → R, j = 1, . . . ,m. (1)
2.2. Surface Averaging
The zero level set of each functionf j in Eq. (1) de-fines a surfaceSj . To obtain the ensemble surfaceS, weneed to average these functions into a single functionfwhose zero level set will defineS.
(a) original surface (b) input points (c) single MPU (d) simple averaging (e) robust averaging
Figure 1. Tangle Cube: Comparison of surfaces obtained from single MPU and ensembles
The choice of the averaging rule usually depends onour knowledge of the functionsf j . If we know the dis-tribution of their errors, we may find an average func-tion which will provably converge to zero error [11, 5].On the other hand, if the properties of the output func-tions are unknown, which is the case of this paper, themean averaging,
f (x) =1m
m
∑j=1
f j(x), (2)
can be the only available option from a theoretical pointof view.
However, in practice, we can improve the robustnessof the ensemble, by removing probable outliers from theset of function values we average. Without loss of gener-ality, we assume that at pointx we have
f1(x)≤ f2(x)≤ . . .≤ fm(x). (3)
Then we compute the average atx as
f (x) =1
m−2r
m−r
∑j=r+1
f j(x). (4)
In other words, we compute a mean average of the func-tions f j atx after excluding ther smallest and ther largestvalues. We user = bm/4c, meaning that we use about ahalf of the values that are more reliable. As one would ex-pect, the average in Eq. (4) is more efficient in handlingoutlier noise.
The function f defined by the averaging in Eq. (4)is continuous. Indeed, any possible discontinuities couldonly appear when the ordering of the functionsf j givenin Eq. (3) changes. However, thef j ’s are themselves con-tinuous functions and so their ordering can only changeat points where they have the same value. This meansthat the average in Eq. (4) will also be continuous. To ob-tain the models shown in Section 4, we evaluate the av-erage functionf at the nodes of a regular grid and thenuse the marching cube algorithm [9] to extract a trian-gle mesh approximating the zero level set off .
3. Validation Test
To validate the ensemble algorithm, we tested it on asurface with a known analytic formula. We used thetan-gle cubegiven by the equation
x4−5x2 +y4−5y2 +z4−5z2 +11.8 = 0, (5)
which has a fairly complex shape and non-trivial topol-ogy (see Fig. 1(a)).
In the set up for the validation test, we follow the ap-proach used in [7]. After sampling a large number ofpoints from the tangle cube, hereN = 244,936 points,we add 30% more points with noise. The noise is a ran-dom spatial displacement from the original position oflength less than 7% of the diagonal of the bounding box(see Fig. 1(b)). The normals are estimated using the clas-sic method proposed in [6].
Figs. 1(c)-(e) show the reconstructed surfaces from thesingle MPU, simple ensemble with Eq. (2), and robust en-semble with Eq. (4), respectively. We measured the RMSerrors of the reconstructions against the original surfacewith the Metro tool [4]. The errors are 0.00789, 0.0055,and 0.00358 for Figs. 1(c)-(e), respectively, which veri-fies the better quality of surface ensembles.
4. Experimental Results and Discussion
Figs. 3 and 4 show the input data and reconstructionsof the Buddha and the armadillo models from the Stan-ford 3D scanning repository. Fig. 2 shows the elevenmembers of the Buddha MPU ensemble. Note that theinput of this experiment is original raw data and the re-sults should not be compared with reconstructions usingfiltered clean data. For the Buddha model, we did not es-timate the normals but used the ones given with the pointset. For the armadillo model, we used [6] to estimate nor-mals. In both cases we did not use the confidence valuesprovided with the data.
On a 3.2 GHz Pentium 4 processor with 2 GB memory,3,260,401 points of the Buddha model were processed bya single MPU in 146 seconds, while the running time of
Figure 2. Buddha: The eleven members of the surface ensemble for reconstructions in Fig. 3.
Figure 3. Buddha: from left to right, pointswith normals, single MPU, and ensemble.
the ensemble was 250 seconds. For the 1,394,271 pointsof the armadillo model, the corresponding times were 92and 304 seconds.
Figure 4. Clockwise from top left: pointswith normals, single MPU, and ensemble
The main disadvantage of the ensemble technique iscomputational overhead. We believe that this trade-offamong speed and robustness is justified in surface recon-struction given that the process is not yet fully automatedand the extra spent computation can save human labor.
Acknowledgements
The authors would like to thank Yutaka Ohtake forgreat help on implementing MPU ensembles. The Bud-dha and armadillo models are courtesy of the StanfordComputer Graphics Lab. This research was supported inpart by the BK21 program, the ITRC support program,and KOSEF (F01-2005-000-10377-0) in Korea and theEuropean Union’s Aim@Shape Network of Excellence.
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